This is the fourth Farey sequence

which is the list, written in increasing order, of all the fractions with values between 0 and 1 that use only the numbers 0,1, 2, 3, 4 as numerators and denominators.

Now if we use the number 5 as well we get the fifth Farey sequence:

Write down all the Farey sequences F ₁ ,F ₂ ,...,F ₁₀ .

Sometimes people make a mistake when adding fractions and, instead of finding the sum, they find the mediant. For example the sum of ¹/₂ and ¹/₃ is ⁵/₆ but the mediant (which you get by adding the numerators and denominators) of ¹/₂ and ¹/₃ is ²/₅. If you add two positive fractions the sum is bigger than both of them but the mediant lies between them. .

Here is the fourth Farey sequence in red with some mediants written in green in the list.

What do you notice? How do the fourth and fifth Farey sequences relate to each other? Does the same relationship hold between the other Farey sequences that you have found?